Computer-generated filtering method

ABSTRACT

A method of optical filtering in which a phase object is constructed in accordance with the desired convolutional filter operator and during the filtering operation incoherent light is used. This type of incoherent filtering method is made applicable to filter operators having negative portions which would require negative intensities by biasing the phase object to make is everywhere nonnegative. The bias is then removed during display by means of a vidicon system.

United States Patent Hirsch et al.

[ Feb. 15, 1972 [54] COMPUTER-GENERATED FILTERING METHOD [72] Inventors:Peter M. Hirsch; James A. Jordan, Jr.;

Louis B. Lesem, all of Houston, Tex.

[73] Assignee: International Business Machines Corporation, Arrnonk,N.Y.

[22] Filed: Nov. 26, 1969 [21 Appl. No.: 880,260

[52] U.S. Cl ..3S0/l62 SF, 235/181 [51] Int. Cl ..G06g 9/00, G02b 27/38[58] Field ofSearch ..350/3.5, 162 SF; l78/6.8;

[56] Reierenees Cited UNITED STATES PATENTS Robbins ..350/ 162 OTHER PUBLICATIONS Trabka, et al., Journal of the Optical Society of America.

Vol. 54,No. 10, Oct. 1964,pp. 1242-1252 Lohmann, Applied Optics, Vol. 7,No. 3, March 1968. pp. 561-563 Lesem, et al., Communications of the ACM,Vol. 11. No. l ;..Q .i- 2 BB:!

Primary Examiner-David Schonberg Assistant Examiner-Ronald J. SternAttorney-Hanifm and Jancin and John L. Jackson ABSTRACT A method ofoptical filtering in which a phase object is constructed in accordancewith the desired convolutional filter operator and during the filteringoperation incoherent light is used. This type of incoherent filteringmethod is made applicable to filter operators having negative portionswhich would require negative intensities by biasing the phase object tomake is everywhere nonnegative. The bias is then removed during displayby means of a vidicon system.

7 Claims, 7 Drawing Figures PATENTEUFEB 15 I972 3. 642.350

SHEET 1 or z AL FIRST CENTRAL ORDER PRIOR ART \fgm FIRST DER IMAGE FIG.I i

INVENTORS PETER H HIRSCH Y JAMES A. JORDAN,JR. LOUIS a. LESEM ATTORNEYmmenm 15 1912 3. 642-; 350

sum 2 0F 2 SUMMER 1 EL (X,Y,Z

TORS

PETER SCH JAMES A JORDAN,JR.

LOUIS B. LESEM ATTORNEY BY 6W I COMPUTER-GENERATED FILTERING METHODBACKGROUND OF THE INVENTION 1. Field of the Invention This inventionrelates to optical information processing and filtering in general, andmore particularly, to that of computer-generated optical filters for usein an incoherent filtering system.

2. Description of the Prior Art Optical information processing is ascience of image processing using stops and diffraction patterns. Manymathematical procedures such as multiplication, correlation, etc., arepossible using holographic diffraction patterns in a coherent opticalsystem. Correlation can also be achieved in incoherent systems. Theseoperations, together with others such as inverse filtering can beaccomplished using computergenerated or synthetic holographic filters inthe coherent optical system. One holographic filtering system isdescribed in Applied Optics, Vol. 7, No. 3, Mar., 1968 at page 561.

An excellent text treatment is also presented in Introduction to FourierOptics, McGraw-Hill, by Joseph W. Goodman, Chapter 7Spatial Filteringand Optical Information Processing.

The usual holographic optical filtering practice makes use of a laser asa coherent light source, several complicated optical elements, theholographic diffraction pattern, and a detection scheme. Inherent inholographic diffraction patterns are two or more diffraction orders.These may be separated angularly in a two-beam hologram. If they are notseparated, the desired diffraction order is obscured by the undesiredorders. If they are separated, the desired order is diffracted away atthe expense of bandwidth from the optical axis of the laser. Severalproblems are presented by these conventional optical informationprocessing systems. These include inefficient utilization of theavailable light since very little of the light from the object to befiltered is diffracted by the hologram into the desired order. Also, therequirement that orders be separated limits the spatial location of theimage, which in turn limits the size and/or resolution of the object tobe filtered. Additionally, such systems are usually costly and complexand, due to the extreme rigidity requirements of coherent imagingsystems, require an optical bench. It is also well known that noiseproblems are also present in any coherent system. Noise may arise fromdust and speckling or diffraction.

Finally, the applications available to coherent filtering systems areseverely limited in that due to the requirement of coherent illuminationreal-time" processing is virtually impossible. That is, if coherentsystems are used, the image must be illuminated coherently. Data, whichmight be in electronic form, for example, must be converted anddisplayed in such a way that a photographic transparency can be made andthis transparency illuminated. This step precludes real-time dataprocessing.

Almost all of the problems associated with coherent, holographic opticalprocessing systems can be eliminated if kinoforms are used. If kinoformsare used in incoherent systems, matched filtering or correlation may beaccomplished.

The kinoform process is described in US. Pat. application Ser. No.778,525, entitled The Kinoform: Method of Manufacturing Wave ShapingDevices," by the inventors of the present invention and assigned to thesame assignee as the present application. In addition, the kinoformprocess was described in a paper presented to the Optical Society ofAmerica Meeting on Mar. 13, 1969 and this paper is published in Volume13, N0. 2 of the IBM Journal of Research and Development, page 150 etseq.

The kinoform is a wave front reconstruction device which, like thehologram, provides the display of a three-dimensional image. In contrastto the hologram, however, the illuminated kinoform yields a singlediffraction order and ideally, all the incident light is used toreconstruct the one image. All the spareqq tial frequency content orbandwidth of the device is available for the single image.computationally, kinoform construction is faster than hologramconstruction because reference beam and image separation calculationsare unnecessary.

A kinoform operates only on the phrase of an incident wave front, beingbased on the assumption that only the phase information in a scatteredwave front is required for the construction of an image of thescattering object. The amplitude of the wave front in the kinoform planeis assumed constant as is approximately true for any diffuselyscattering object in the far field. Although it was first conceived asan optical focusing element, the kinoform can be used to transform thewave front of any physical waveform; e.g., ultrasound or microwaves.

The ability to transform a wave front at will permits the use of thekinoform to represent a class of mathematicaloperators. These operatorsare real, nonnegative, convolutional operators, ofwhich the correlationoperatorsor matched filter are a particular example.

Because of the random phase assumption made in the construction ofkinoforms, they cannot be used in coherent optical processing systems.However, kinoforms are ideally suited for use in incoherent systems. Aswith holographic filters operating in incoherent systems, the operationsare performed on intensities, rather than amplitudes. Thus, basickinoformic systems are applicable to operations which are real andnonnegative, such as matched filtering. The basic kinoform filteringsystem is the subject of a patent application entitled Kinoform MatchedFilter Method by Adolph W. Lohmann, Ser. No. 880,258 filed on the sameday as the present application and assigned to a common assignee.

Other related applications are A Method for Figuring Lenses, Ser. No.813,641 and Discrete Aperture Method of Making Synthetic Kinoforms andHolograms, Ser. No. 794,977, both by the inventors of the presentinvention and assigned to the assignee of this application.

Thus, while basic kinoform filtering systems overcome most of theproblems discussed above which are associated with coherent systems,i.e., cost, complexity, noise, bandwidth, and no real-time processing,the available applications are limited to real, positive operators.Since many standard processing techniques require the use of operatorswhich may be negative or even complex, the extension of incoherentfiltering systems to these types of operators is highly desirable.

SUMMARY OF THE INVENTION Briefly, a filter operator to be implemented ismathematically defined. The operator may, for instance, be a velocityfan filter for use in analyzing seismic traces. This type of filter iswell known in the seismic data manipulation art. One fan filtertechnique was described in a paper presented at the thirtyeighth Societyof Exploration Geophysicists Convention in Denver, Colorado on Oct. 7,I968 by J. C. Patau. As there described, the filter is represented by amatrix of values and this matrix is digitally convolved on the digitizedtraces to be processed.

In the present invention in the preferred embodiment, the matrix ofvalues is processed and a constant added such that all of the values arereal and nonnegative and the required phase to represent this matrixcalculated and plotted. The plot is then photoreduced and bleached toprovide the kinoform filter. The phase calculation, plotting andbleaching are as described in the aforementioned patent application,Ser. No. 778,525.

During actual filtering the object to be filtered such as a transparencyhaving seismic traces thereon is illuminated with incoherent light whichpreferably is color filtered and diffused. The light scattered by theobject then passes through the filter and the resultant filtered image,imaged by means of a lens.

Other alternate embodiments including one in which the incoherentlyilluminated object to be filtered is scanned through the filter anddisplayed on a cathode-ray tube such that real time processing isaccomplished, are also provided.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is an illustration of atypicalholographic coherent information-processing system;

FIG. 2 is an illustration of one embodiment of the subject novelincoherent kinoform information-processing system;

FIG. 3 is an illustrative linear mathematical operator, the impulseresponse of a fan filter;

FIG. 4 is a view illustrative of the preferred systems embodiment of thesubject invention;

FIG. 5 is an alternate embodiment in which both positive and negativefilters are used to produce a convolutional filter;

FIG. 6 is an optical system which could be used to accomplish opticallythe electronic summing of the system of FIG. 5;

FIG. 7 is an optical system which could be used in the system of FIG. 5for processing complex numbered operators.

DETAILED DESCRIPTION OF THE DRAWINGS To aid in an appreciation andunderstanding of the subject novel technique, refer first to FIG. I,where there is an illustration of a coherent optical processing system.using holographic members. While this system is described in detail inthe aforementioned Applied Optics reference and Goodman text, a briefdescription will be given to aid in an appreciation of the presentincoherent filtering system. In FIG. I, a light source 1 such as a laserprovides a coherent, monochromatic wave front which is shaped into aplane or spherical wave front 5 by an objective lens 2, spatial filter 3and collimating lens 4. The wave front 5 illuminates the object 6 whichcontains data which is to be filtered; i.e., a convolutional operationis to be carried out on the one-, two-, or three-dimensional data. Asillustrated in FIG. 1, the data is carried on a transparency such that acoherent wave front is scattered from it. The wave front 7 illuminatesthe holographic member 8. Most of the light is transmitted in thecentral diffraction order, while relatively little is transmitted in thefirst order. The desired filtered image appears in the real first order.

As briefly discussed above, this type of filtering or processing systemhas many attendant disadvantages such as high cost and complexity.Additionally, for many applications it is made impractical by itsrequirement, due to the use of coherent light, of very accuratealignment and extreme stability which necessitate an optical bench and askilled craftsman. Also, the requirement that orders be separated limitsthe spatial location of the image, which in turn limits the size and/orresolution of the object (data) to be filtered. Finally, not only isthere an inefficient use of light in that most of the incident light isdiffracted into the central order, but additionally, realtime processingis precluded due to the requirement that the data to be filtered must bereduced to photographic form such that it can be caused to transmitcoherent illumination.

Several attempts have been made in the past to make holographicfiltering more practical. Since all of the above listed problems arisefrom the use of coherent light, attempts have been made to constructincoherent systems. These systems have approximated coherent light byviewing only a small area and have been unsatisfactory due to poor imagequality.

Many of the above problems are overcome by the basic kinoform filteringtechnique which is the subject of the aforementioned patent application.In FIG. 2 is illustrated the basic kinoformic incoherent filteringsystem. As illustrated, the transparency 9 containing the data to befiltered is illuminated by quasi monochromatic, temporally incoherentwaves I0.v The waves [0 emanate from, for instance, an incandescentlight 11 and prior to their arrival at the object 9 are color filteredby filter 12, such that only waves of the spectrum for which thekinoform filter 16 was designed are passed. The waves are also passedthrough a diffuser 13.

The object 9 can be thought of being made up ofmany point sources, witheach point source of a specified intensity and with a phase varying withtime, i.e., temporally incoherent. The points 14 and 15 represent twosuch point sources. Each point source 14, 15 illuminates the kinoform 16which in turn produces a virtual image 17, 18 positioned as shownrelative incoherent light, the total intensity at point (a,b,z) is givenby v the equation If a lens 19 is used to image this virtual patternonto a screen or other image recording device 20, then the patternbecomes These two cases can be thought of as correlation orconvolutional filtered objects. This system has numerous advantages suchas it is relatively low in cost and is not complex in that it hasneither the extreme alignment nor imaging requirements of a coherentsystem. Additionally, there is no noise problem from microscopic dustparticles or flaws in the optics since these are averaged out in anincoherent system. Furthermore, real-time processing can be accomplishedsince it is possible to use real-time data. That is, the data to beprocessed can be displayed on a cathode-ray tube and the light from theCRT phosphor used as the illumination. Also, since it is a single-ordersystem there is no overlapping order problem which limits the size ofthe object to be filtered and therefore very large objects can befiltered using very small filters.

Although this system has great advantages for optical informationprocessing, as above discussed, its applications are limited by the factthat the function F consists of intensities which are nonnegative.Negative intensitieshave no meaning, and thus operators which are inpart negative cannot be used in the system of FIG. 2.

It should be understood in connection with FIGS. 2, 4-6 that the termincoherent light is used in its true sense. That is, while asillustrated in these figures, for purposes of quality, a diffuser andcolor filter are used along with a separate" source of incoherent light,these members are not required. The ordinary ambient light reflectedfrom a piece of paper having the data to be filtered printed on it issufficient. In this type of simple set up the data is viewed through thekinoform filter and the eye constitutes the lens or assuming that theambient light is great enough, the filtered image is scanned through thefilter. I

Further with respect to color filtering, while in good quality systemsthis is desirable, it is not necessary. That is, the kinoform filterswhich have actually been constructed have been "tuned for use with redlight. These filters have been satisfactorily used in systems where theonly light is reflected light from normal fluorescent lights and withillumination from a black-white CRT display.

The novel invention described hereinafter provides several techniquesfor using the very great simplification inherent in the kinoformicoptical processing system of FIG. 2 for all convolutional operatorsregardless of whether the operator values are positive or negative, aswell as for certain nonlinear ones. These novel techniques allow for theapplication of one-, two-, or three-dimensional operators to one-, two-,or three-dimensional objects (data sets). Because the kinoform can beused for nonoptical waves, e.g., nonvisible electromagnetic waves, soundwaves, information-processing systems using those media may also beconstructed using techniques analogous to those herein described.

To facilitate a description of the present novel technique, consider theoperator illustrated in FIG. 3. This operator is a single example of alarge class of mathematical convolutional operators. Some examples ofconvolutional operators are matched operators, inverse operators,cross-correlation operators, deconvolution operators, fan-filteringoperators, autocorrelation operators, band-pass operators, high-passoperators, notch operators, reject operators, and low-pass operators.

Many of these examples have negative regions, which cannot berepresented directly in a kinoform. Some have both negative and complexregions, which present even greater problems.

The novel invention described herein provides a technique of using akinoformic incoherent optical processing system for processing using aconvolutional filter limited to three or fewer dimensions.

In the first embodiment, a general real convolutional operator, forexample the fan-filtering operator F(x,y,z) illustrated in FIG. 3, canbe applied optically, using a kinoform. In this case the operator hassome minimum value which will be referred to as C. If a constant numberC is added to the operator F(x.y.z), the resulting operatorF(x,y,z)=F(x,y,z)+C is everywhere nonnegative. If a kinoform is producedin which the amplitude of the virtual image is given by the function(F'(.r,y,z))" it can be used in the system illustrated in FIG. 2. Thesquare rooting is required since the square of the amplitude equals theintensity. If the amplitude of the light passed through the object isgiven by 6(x,y,z) then the intensity distribution at a detector screenis G(a,b,c), the processed image can also be written as |0(x, y, z)|F(ax. by, c-z) JUZ The first term is the desired convolution of 0 (x,y,zwith the operator F (ax,by,cz). The second term is a constant. If thesecond term is called K,

2 2 W y z) I then this constant K can best be removed from the observedimage G(a,b,c) in real time by placing a vidicon tube or other TV typecamera tube in scanning alignment with the kinoform and byelectronically subtracting a voltage equal to that produced by theillumination K. Biasing techniques for voltage subtraction are common inelectronics technology.

Thus, the difference between the basic kinoform filtering system and thepresent system is twofold. The first difference is the manner ofcalculating the filter, i.e., the addition of the constant and thesecond difference is in the manner of utilization of the filtered data.That is, with respect to the second difference, the bias or constantadded in most cases must be removed. To accomplish removal of this bias,the preferred manner is by electronic means. Thus, the lens 19 of FIG. 2would in this case represent the lens of a scanner such as a vidicon andthe resultant, image displayed at 20 which would constitute the face ofa display tube such as a CRT. Such a system is illustrated in FIG. 4 inwhich as shown the light source 11, color filter l2, diffuser 13, datamember 9 and kinoform filter 16 are operative to produce a virtual image17 and 18 which is scanned by vidicon 22 which has its output biasadjusted and amplified at 23 for display at 24.

Removal of the constant term K can also be accomplished using verycarefully controlled photographic techniques in which the nonlinearitiesof the emulsion and developing method are used to suppress the constantsignal. Obviously this variation of the first embodiment precludesreal-time data processing and would require a skilled photographer,whereas the vidicon variation requires little skill of the operator.

Again, in connection with FIG. 4, as above mentioned, the data member 9could be either a photographic transparency or could be the face ofadisplay tube. In the event that 9 were the face of a display tube, colorfiltering, if desired, could be accomplished by the choice of phosphors.The light source 11 would be that furnished by the display and a goodamount of diffusion is inherent in CRT displays.

Filter 35 =P(x, y, z) {F(x, 3/, z) where F(x, y. z) 2 0 0 elsewhereFilter 36=1V(x, y, z) {-F(x. y, z) where F(x. y. z) (l v flels her Notethat The results are combined either optically or electronically at 26.As illustrated, the light transmitted by each filter 35 and 36 isdetected by a vidicon 38 and 39, respectively, and the summed resultdisplayed at 40. This process is [6(x, y, Z)

Again, as mentioned with respect to the other embodiments, the data tobe filtered could be displayed, for real time processing on a CRTdisplay.

In FIG. 6 there is shown an illustration of an optical system whichcould be used to accomplish the summing. In this embodimentmonochromatic light and equal path lengths would have to be used toassure accuracy in the subtraction. Thus, as illustrated symbolically,the data-scattered incoherent light 42 from data member 41 is dividedinto two beams 43 and 44 by a beam splitter 45. Beam 44 passes throughthe positive filter 46 and thence through a half-silvered mirror whereit is combined with beams 43 which has been reflected by a mirror 48,through the negative filter 49 which produces N(x,y,z), through thehalf-wave plate 50 to produce N(x,y,z) which is then reflected frommirrors 51 and 47 to produce F (x,y,z) as indicated by the arrow.

This is exactly the result desired of convolving the intensities of theimage with the general convolution operator.

It is also possible to produce complex operators by using four filters,two filters as mentioned for the real part of the function and twoadditional filters Pl(x,y.z) and Wl(x.y z) for the imaginary parts ofthe filter. These filters can be combined optically to produce thedesired operator. Mathematically, h sen be ssssribsaa Jl/Z Optically,this can be implemented as illustrated in FIG. 7. There, data-scatteredincoherent light 55 from data member 56 is split into two beams 60 and61 by beam splitter 62. Beam 60 then passes through quarter wave plate63 to provide the data-scattered light 64 for filter 65 which is thepositive complex filter. Beam 61 is reflected from the mirror andprovides the data-scattered light for the positive filter 67.

,t r as $9352. F '11P? heY fl by Beam 58 is reflected from mirror 68,through half-wave plate 69 and is split into two beams 70 and 71 by beamsplitter 72. Beam 70 provides the data-scattered light for filter 73which is the negative filter. Beam 71 is reflected by mirror 74 andafter passing through quarter wave plate 75 provides the data-scatteredlight for filter 75.

The output from the optical system would be obtained by combining thelight from filters 65, 67, 73, and 75 in a manner opposite to that justdescribed for the input.

In summary and to tie in the aforereferenced kinoform technique which isdescribed in patent application Ser. No. 778,525, a discussion of thecalculation and construction of a filter will be provided.

During calculation of the filter, the impulse response function isconsidered to be a three-dimensional array of point apertures. Eachaperture is assigned a value between zero and one, where zero impliesthat no light is transmitted through the aperture, one implies an openaperture, and the values between represent the relative transmittance ofthe apertures. These values are made to correspond to the square root ofF(x,y.z) or F (x,y,z), depending upon the embodiment. These values areread into a calculating machine, using, for example, punched cards, anda plot tape is generated. In the remainder of this discussion, the firstembodiment and a one-dimensional treatment will be given for simplicity.

The first s'tepin geiieratr rig th'e 'pIo't tapers amusements impulseresponse function into a vector of m elements to mula phase factor M r arandom; i dlqi lifi ldifi T Ihenlgtt na Bail ififlifi the next step isto use the discrete form of the Kirchhoff diffraction formula tocalculate the wave front at the kinoform filter position required toreproduce the impulse response function F. in the Fresnel approximation,this is accomplished by calculating In the calculations, zeros areappended to the F array so that it is a vector of n elements. Thisinterpolates the TE (transform) array:

n/2-l. [has the range from -n/2 to (n/2 )I.

Since the TE array is of period n, it may be repeated as many times asnecessary, to provide a filter as large as desired. The TE array has theform of In the generation of the kinoform filter, only the phase w(l/p),mod 2mr is used; the amplitude A(I/p) is assumed to be constant.

The introduction of the phase factor exp [il (a,b)], which simulates theground glass or the point aperture format alleviates the need forconsidering amplitude in the calculation.

The phase co(l/p) is plotted on a plotter with, for instance, 32 graylevels, such that the phase ranges from Oto 211' over the scale. Theplot is then photoreduced to the appropriate size, governed by thewavelength of light used, and the design distance from the data to befiltered to the filter. The photoreduced device is then etched, forexample, with Kodak etch bath EB-3. The etch bath etches the surface ofthe photoreduction in proportion to the darkening of the photographicreduction. The etching of the photoreduction for a kinoform filter mustbe performed with much more care than is required for conventionalbleached holograms. The relief of the emulsion must be such that lightincident upon a region of l will be retarded by one wavelength, comparedwith the light incident upon a region of =21r. When phase matching isachieved, almost all of the light incident upon the kinoform filter willpresent in the desired impulse response function with no spuriousorders.

While the subject invention has been described systemwise with thefilter being photographically produced, due to the noncriticality of therequired light, both the filter and the data could be real-timedisplayed. This could be accomplished with two deformographic storagedisplay tubes, one for displaying the object to be filtered and a secondto display the desired kinoform. The displays could come from a vidiconor a computer to allow real-time filtering.

Additionally, it will be obvious to those skilled in the art that thesubject invention is equally applicable to other than the optical-typeapplications herein described. Thus, for instance, sonic and ultrasonicfilters could be readily implemented. In this event, however, as will beobvious the filter, while being calculated in exactly the same manner asherein described, would be made of different materials, depending on theapplication. Therefore, other techniques, such as cutting and milling,rather than bleaching would be employed.

While the invention has been particularly shown and described withreference to several embodiments, it will be understood by those skilledin the art that various changes in form and detail may be made withoutdeparting from the spirit and scope of the invention.

What is claimed is:

1. A method of performing the mathematic convolution between athree-dimensional convolutional operator and a three-dimensionalfunction by processing physical incoherent waves emanating from aphysical amplitude distribution with a processing member to produce azero diffraction order output at an output plane comprising the stepsof:

A. representing said function by a real physical object which whenilluminated by a wave front scatters said wave front according to saidamplitude distribution to provide said physical incoherent waves;

B. discretizing said convolutional operator across a threedimensionalmatrix to obtain a matrix of positive values;

C. effectively removing any negative value from said matrix of values byadding a positive constant to all of said values to form a matrix ofnonnegative values;

D. constructing said processing member according to the processes formaking a kinoform wherein said discretized convolutional operator isconsidered to be the intensity of the image projected by said kinoform;

E. illuminating said processing member with said physical incoherentwaves to selectively retard said waves with resultant interference atsaid output plane in the zero diffraction order corresponding to saidconvolution; and

F. compensating for said positive constant added in step (C) bysubtraction of a bias from said convolution at said output plane.

2. The method of claim 1 wherein said physical incoherent waves areincoherent light waves such that optical distribution of energy occursat said output plane.

3. The method of claim 2 wherein the resultant distribution of energy atsaid output plane is scanned by a vidicon for sub sequent display andprior to display a negative bias corresponding to the constant added tosaid matrix of values is added. a

4. The method of claim 3 wherein said processing member is made of amaterial of substantially uniform transmissivity but with selectivelyvaried thickness corresponding to said controlled phase retarding areas.

5. The method of claim 4 wherein said selectively varied thickness isobtained by calculating the phase distribution required to produce saidoperator in optical form at said output plane with the assumption thatsaid physical incoherent light waves are from a point source, plottingsaid calculated phase distribution as amplitude on a multigrey levelplotter, photoreducing said plot and bleaching said photoreduction.

1. A method of performing the mathematic convolution between athree-dimensional convolutional operator and a three-dimensionalfunction by processing physical incoherent waves emanating from aphysical amplitude distribution with a processing member to produce azero diffraction order output at an output plane comprising the stepsof: A. representing said function by a real physical object which whenilluminated by a wave front scatters said wave front according to saidamplitude distribution to provide said physical incoherent waves; B.discretizing said convolutional operator across a threedimensionalmatrix to obtain a matrix of positive values; C. effectively removingany negative value from said matrix of values by adding a positiveconstant to all of said values to form a matrix of nonnegative values;D. constructing said processing member according to the processes formaking a kinoform wherein said discretized convolutional operator isconsidered to be the intensity of the image projected by said kinoform;E. illuminating said processing member with said physical incoherentwaves to selectively retard said waves with resultant interference atsaid output plane in the zero diffraction order corresponding to saidconvolution; and F. compensating for said positive constant added instep (C) by subtraction of a bias from said convolution at said outputplane.
 2. The method of claim 1 wherein said physical incoherent wavesare incoherent light waves such that optical distribution of energyoccurs at Said output plane.
 3. The method of claim 2 wherein theresultant distribution of energy at said output plane is scanned by avidicon for subsequent display and prior to display a negative biascorresponding to the constant added to said matrix of values is added.4. The method of claim 3 wherein said processing member is made of amaterial of substantially uniform transmissivity but with selectivelyvaried thickness corresponding to said controlled phase retarding areas.5. The method of claim 4 wherein said selectively varied thickness isobtained by calculating the phase distribution required to produce saidoperator in optical form at said output plane with the assumption thatsaid physical incoherent light waves are from a point source, plottingsaid calculated phase distribution as amplitude on a multigrey levelplotter, photoreducing said plot and bleaching said photoreduction. 6.The method of claim 5 wherein said physical incoherent light waves areprovided by a cathode-ray tube having displayed thereon said data to beprocessed.
 7. The method of claim 5 wherein said physical incoherentwaves are provided by illuminating said data in visual form withincoherent light.